# zbMATH — the first resource for mathematics

Explicit realization of a metaplectic representation. (English) Zbl 0727.11021
Let F be a local field which is neither $${\mathbb{C}}$$ nor has residual characteristic 2. Then there exists a 2-fold covering group of $$GL_ r(F)$$ analogous to the metaplectic group of Weil. There exists also a class of remarkable representations of this group called the $$\Theta$$- representations. These have been constructed as subquotients of principal series representations. The purpose of this paper is to give a realization of them on a space of functions for the case $$r=3$$. This involves giving a special linear form (let us call it L) on each of the representations and determining the space spanned by the functions $$g\mapsto L(gv)$$ as v runs through the space of the representation. In this case the authors construct the representation on a space of functions on a 2-fold cover of $$F^ 2$$-$$\{$$ $$0\}$$. The existence of the model is deduced from the structure of the space of Whittaker functionals on $$\Theta$$-representations of the 2-fold covers of $$GL_ 2(F)$$ and $$GL_ 3(F)$$ by means of the Bernstein-Zelevinski extension of Gelfand- Kazhdan theory.

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields 11F37 Forms of half-integer weight; nonholomorphic modular forms
Full Text:
##### References:
 [1] I. Bernstein and A. Zelevinsky,Representations of the group GL(n, F)where F is a non-archimedean local field, Russ. Math. Surv.31 (1976), 1–68. · Zbl 0348.43007 [2] I. Bernstein and A. Zelevinski,Induced representations of reductive groups I, Ann. Sci. Ec. Norm. Super.10 (1977), 441–472. · Zbl 0412.22015 [3] W. Casselman,Characters and Jacquet modules, Math. Ann.230 (1977), 101–105. · Zbl 0337.22019 [4] Y. Flicker,Automorphic forms on covering groups of GL(2), Invent. Math.57 (1980), 119–182. · Zbl 0431.10014 [5] Y. Flicker,Explicit realization of a higher metaplectic representation, Indag. Math. (1990), to appear. · Zbl 0725.22008 [6] Y. Flicker and D. Kazhdan,Metaplectic correspondence, Publ. Math. IHES64 (1987), 53–110. [7] Y. Flicker and D. Kazhdan,On the symmetric square: Unstable local transfer, Invent. Math.91 (1988), 493–504. · Zbl 0637.10022 [8] Y. Flicker and J. G. M. Mars,Summation formulae, automorphic realizations and a special value of Eisenstein series, J. Number Theory (1990), to appear. · Zbl 0788.11017 [9] R. Howe and C. Moore,Asymptotic properties of unitary representations, J. Funct. Anal.32 (1979), 72–96. · Zbl 0404.22015 [10] A. Kirillov,Elements of the Theory of Representations, Grundlehren 220, Springer-Verlag, Berlin, 1976. · Zbl 0342.22001 [11] D. Kazhdan and S. J. Patterson,Metaplectic forms, Publ. Math. IHES59 (1984), 35–142. · Zbl 0559.10026 [12] D. Kazhdan and S. J. Patterson,Towards a generalized Shimura correspondence, Adv. in Math.60 (1986), 161–234. · Zbl 0616.10023 [13] J. Milnor,An introduction to algebraic K-theory, Ann. Math. Stud.72 (1971). · Zbl 0237.18005 [14] P. Torasso, Quantification géométrique, opérateurs d’entrelacements et représentations unitaires de SL(3, R), Acta Math.150 (1983), 153–242; voir aussi:Quantification géométrique et représentations de SL3(R), C. R. Acad. Sci. Paris291 (1980), 185–188. [15] A. Weil,Sur certains groupes d’opérateurs unitaires, Acta Math.111 (1964), 143–211. · Zbl 0203.03305
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.